ECMLPKDD 2010 Barcelona Spain B Aditya Prakash Hanghang Tong Nicholas Valler Michalis Faloutsos Christos Faloutsos Carnegie Mellon University Pittsburgh USA ID: 437353
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Slide1
Virus Propagation on Time-Varying Networks: Theory and Immunization Algorithms
ECML-PKDD 2010, Barcelona, Spain
B. Aditya Prakash*, Hanghang Tong* ^, Nicholas Valler+, Michalis Faloutsos+, Christos Faloutsos*
*
Carnegie Mellon University, Pittsburgh USA
+
University of California – Riverside USA
^
IBM Research,
Hawthrone
USASlide2
Two fundamental questions
Epidemic!
Strong Virus
Q1: Threshold?Slide3
example (static graph)
Weak Virus
Small infection
Q1: Threshold?Slide4
Questions…
Q2: Immunization
Which nodes to immunize?
?
?Slide5
Standard, static graphSimple stochastic framework
Virus is ‘Flu-like’ (‘SIS’)Underlying contact-network – ‘who-can-infect-whom’Nodes (people/computers) Edges (
links between nodes)OUR CASE:Changes in time – alternating behaviors!think day vs night Our FrameworkSlide6
‘S’ Susceptible (= healthy); ‘I’ InfectedNo immunity (cured nodes -> ‘S’)
Reminder: ‘Flu-like’ (
SIS)SusceptibleInfected
Infected by neighbor
Cured internallySlide7
Virus birth rate β
Host cure rate δSIS model (continued)
Infected
Healthy
X
N1
N3
N2
Prob.
β
Prob.
β
Prob.
δSlide8
Alternating Behaviors
DAY (e.g., work)
adjacency matrix
8
8Slide9
Alternating Behaviors
NIGHT
(e.g., home)
adjacency matrix
8
8Slide10
√Our Framework
√SIS epidemic model√Time varying graphs Problem Descriptions
Epidemic Threshold Immunization ConclusionOutlineSlide11
SIS modelcure rate δ
infection rate β
Set of T arbitrary graphs Formally, given
day
N
N
night
N
N
….weekend…..
Infected
Healthy
X
N1
N3
N2
Prob.
β
Prob.
β
Prob.
δSlide12
Find…
Q
1: Epidemic Threshold:Fast die-out?Q2: Immunizationbest k?
?
?
above
below
I
tSlide13
NO epidemic if
eig (S) = < 1
Q1: Threshold - Main resultSingle number! Largest eigenvalue of the “system matrix ”Slide14
NO
epidemic if
eig (S) = < 1S =
cure rate
infection rate
……..
adjacency matrix
N
N
day
night
DetailsSlide15
Synthetic100 nodes - Clique - ChainMIT Reality Mining
104 mobile devicesSeptember 2004 – June 200512-hr adjacency matrices (day) (night)
Q1: Simulation experimentsSlide16
‘Take-off’ plots
Synthetic
MIT Reality MiningFootprint (# infected @ steady state)
Our threshold
Our threshold
(log scale)
NO EPIDEMIC
EPIDEMIC
EPIDEMIC
NO EPIDEMICSlide17
Time-plots
Synthetic
MIT Reality Mininglog(# infected)Time
BELOW threshold
AT threshold
ABOVE threshold
ABOVE threshold
AT threshold
BELOW thresholdSlide18
√Motivation
√Our Framework √SIS epidemic model
√Time varying graphs√Problem Descriptions√Epidemic Threshold Immunization ConclusionOutlineSlide19
Our solutionreduce (== )
goal: max ‘eigendrop’ Δ
Comparison - But : No competing policyWe propose and evaluate many policiesQ2: Immunization Δ = _before - _after
?
?Slide20
Lower is better
Optimal
Greedy-S
Greedy-
DavgASlide21
Time-varying Graphs ,SIS (flu-like) propagation model
√ Q1: Epidemic Threshold - < 1
Only first eigen-value of system matrix!√ Q2: Immunization Policies – max. Δ OptimalGreedy-SGreedy-DavgAetc.Conclusion
….Slide22
B. Aditya Prakash
http://www.cs.cmu.edu/~badityap
Our threshold
Any questions?